library(nnet)
# logit
mod1 <- glm(vs ~ hp + mpg, data = mtcars, family = binomial)
# Multinomial logit
mod2 <- multinom(factor(gear) ~ hp + am, data = mtcars, trace = FALSE)
pred1 <- predict(mod1, se.fit = TRUE, type = "response")
pred2 <- predict(mod2, se.fit = TRUE, type = "probs")I fit a model… now what?
About me
- Québécois
- PhD University of Michigan
- Associate Professor of Political Science, Université de Montréal
- Social Science Research Methods
- Political Economy
- First upload to CRAN: 2009-06-03
Problem 1: Inconsistency
Predicted probabilities with standard errors:
Problem 1: Inconsistency
Logit:
List of 3
$ fit : Named num [1:32] 0.7034 0.7034 0.8841 0.7005 0.0227 ...
..- attr(*, "names")= chr [1:32] "Mazda RX4" "Mazda RX4 Wag" "Datsun 710" "Hornet 4 Drive" ...
$ se.fit : Named num [1:32] 0.1356 0.1356 0.0905 0.1336 0.0431 ...
..- attr(*, "names")= chr [1:32] "Mazda RX4" "Mazda RX4 Wag" "Datsun 710" "Hornet 4 Drive" ...
$ residual.scale: num 1Multinomial logit:
Problem 2: Interpretation
Call:
glm(formula = vs ~ hp + mpg, family = binomial, data = mtcars)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.09164 -0.19536 -0.01377 0.50499 1.18424
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 9.53119 7.03368 1.355 0.1754
hp -0.07234 0.03461 -2.090 0.0366 *
mpg -0.03385 0.18097 -0.187 0.8516
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 43.860 on 31 degrees of freedom
Residual deviance: 16.803 on 29 degrees of freedom
AIC: 22.803
Number of Fisher Scoring iterations: 7Call:
multinom(formula = factor(gear) ~ hp + am, data = mtcars, trace = FALSE)
Coefficients:
(Intercept) hp am
4 5.262061 -0.049919859 8.312471
5 -9.944043 0.005586538 16.891350
Std. Errors:
(Intercept) hp am
4 3.206564 0.02694180 16.665212
5 17.956716 0.04888094 1.771992
Residual Deviance: 20.09822
AIC: 32.09822 What does “log odds” mean in the real world?
Solution: marginaleffects
Interpretation:
- Post-estimation methods to interpret statistical models
Consistency:
- 5 general purpose technologies
- 68 classes of models supported
Alternatives (try them!)
emmeans
- Most popular
- Feature gap is closing (reversing?)
- Fewer models supported
- Feels complicated (to me!)
margins:
- Author has recommended
marginaleffects
modelbased and ggeffects
- Easy
- Great plots
- Wrappers around
emmeans - Less flexible
Try them; they’re great!
So why marginaleffects?
- Simple
- Powerful
- Composable
- Valid
- More models
- Active development (I have tenure and love this stuff.)
Five general purpose technologies
Interpret statistical models with:
- Delta method
- Adjusted predictions
- Contrasts
- Marginal effects
- Marginal means
Demo: 1892 free CSVs on RDatasets
Fair’s Affairs (1969)
Number of extra-marital affairs, children, age, etc.
affairs <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/AER/Affairs.csv")
tail(affairs) X affairs gender age yearsmarried children religiousness education
596 1935 7 male 47 15.0 yes 3 16
597 1938 1 male 22 1.5 yes 1 12
598 1941 7 female 32 10.0 yes 2 18
599 1954 2 male 32 10.0 yes 2 17
600 1959 2 male 22 7.0 yes 3 18
601 9010 1 female 32 15.0 yes 3 14
occupation rating
596 4 2
597 2 5
598 5 4
599 6 5
600 6 2
601 1 5Fair’s Affairs (1969)
Poisson model:
library(marginaleffects)
mod <- glm(
affairs ~ children + gender + yearsmarried,
family = poisson,
data = affairs)
summary(mod)
Call:
glm(formula = affairs ~ children + gender + yearsmarried, family = poisson,
data = affairs)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.144 -1.769 -1.384 -1.169 6.647
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.389798 0.092200 -4.228 2.36e-05 ***
childrenyes 0.038412 0.105335 0.365 0.715
gendermale 0.030381 0.067653 0.449 0.653
yearsmarried 0.076891 0.007682 10.009 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 2925.5 on 600 degrees of freedom
Residual deviance: 2766.5 on 597 degrees of freedom
AIC: 3268.5
Number of Fisher Scoring iterations: 6Delta method
Standard errors for functions of parameters.
Delta method
The childrenyes and gendermale coefficients are close:
(Intercept) childrenyes gendermale yearsmarried
-0.38979791 0.03841220 0.03038081 0.07689095 Tests of equality:
Delta method
term estimate std.error statistic p.value conf.low
1 b2 * exp(b3) = 0.02 0.0195971 0.1085638 0.1805123 0.8567504 -0.1931841
conf.high
1 0.2323783- Non-linear functions of the params
- Standard errors everywhere
- All
marginaleffectsfunctions have ahypothesisargument- Are two predictions/contrasts/mfx equal?
- Cross-group contrasts
Adjusted Predictions
The outcome predicted by a fitted model on a specified scale for a given combination of values of the predictor variables, such as their observed values, their means, or factor levels (“reference grid”).
aka, in-sample predictions, fitted values.
One predicted value per observation, the usual way:
1 2 3 4 5 6 7 8
1.5060531 0.9210563 2.2299485 2.2987358 0.7395237 0.7599813 0.7173943 2.2987358
9 10 11 12 13 14 15 16
2.2299485 0.7834244 2.2987358 0.9866488 2.2987358 0.7599813 0.9210563 2.2299485
17 18 19 20 21 22 23 24
2.2299485 0.7173943 0.7599813 1.5181975 0.7599813 0.7599813 1.5181975 1.5181975
25 26 27 28 29 30 31 32
0.9866488 0.7599813 1.1600217 2.2987358 0.9866488 0.9571244 2.2987358 0.7599813
33 34 35 36 37 38 39 40
0.7208288 2.2299485 0.9866488 0.7599813 2.2299485 0.9210563 0.8141029 0.7173943
41 42 43 44 45 46 47 48
1.5650294 2.2987358 0.6992588 0.9571244 1.2054476 0.9494681 1.2054476 2.2299485
49 50 51 52 53 54 55 56
0.8141029 2.2987358 0.7173943 1.2426320 0.9866488 1.5650294 0.9494681 2.2299485
57 58 59 60 61 62 63 64
0.8141029 2.2299485 0.7173943 1.5650294 2.2299485 1.5650294 0.7173943 0.9494681
65 66 67 68 69 70 71 72
1.1958049 2.2987358 1.5650294 2.2299485 0.7599813 0.7599813 0.9210563 1.5181975
73 74 75 76 77 78 79 80
0.6837339 2.2987358 2.2987358 1.2426320 0.7897418 0.9866488 0.7599813 2.2987358
81 82 83 84 85 86 87 88
0.8141029 2.2987358 0.7173943 0.9494681 0.7897418 0.6992588 2.2299485 0.7599813
89 90 91 92 93 94 95 96
0.7897418 2.2299485 2.2299485 1.5650294 1.5650294 2.2987358 0.7208288 2.2299485
97 98 99 100 101 102 103 104
2.2987358 1.5650294 2.2987358 2.2987358 1.5181975 1.5650294 0.6837339 2.2987358
105 106 107 108 109 110 111 112
2.2299485 0.9866488 1.2426320 2.2987358 2.2987358 0.7599813 1.2054476 2.2299485
113 114 115 116 117 118 119 120
0.7599813 1.5650294 1.1600217 2.2121107 2.2987358 1.5060531 0.7173943 1.2054476
121 122 123 124 125 126 127 128
1.2054476 2.2299485 2.2987358 1.2054476 0.7599813 2.2987358 0.6837339 0.7834244
129 130 131 132 133 134 135 136
0.7834244 0.7834244 1.5181975 2.2987358 0.7599813 0.9494681 1.5181975 2.2299485
137 138 139 140 141 142 143 144
0.7173943 2.2299485 0.9866488 0.9494681 1.5181975 2.2299485 1.2426320 2.2987358
145 146 147 148 149 150 151 152
0.9494681 0.9866488 0.9866488 2.2299485 2.2299485 1.2426320 1.2426320 0.7395237
153 154 155 156 157 158 159 160
0.9866488 2.2987358 0.7684831 2.2987358 2.2299485 2.2299485 1.2426320 0.9866488
161 162 163 164 165 166 167 168
0.9571244 0.9571244 2.2987358 2.2299485 1.5181975 0.7897418 2.2299485 0.7599813
169 170 171 172 173 174 175 176
2.2987358 2.2987358 2.2987358 0.6837339 1.5181975 2.2299485 0.7599813 0.7324256
177 178 179 180 181 182 183 184
0.9210563 1.5181975 1.2054476 2.2299485 2.2299485 0.8141029 0.9571244 1.1600217
185 186 187 188 189 190 191 192
0.7266414 1.2054476 0.7395237 0.9866488 1.5650294 2.2987358 0.7395237 1.2054476
193 194 195 196 197 198 199 200
0.7490561 2.2987358 2.2299485 0.9866488 2.2987358 0.7599813 2.2987358 0.7599813
201 202 203 204 205 206 207 208
0.9866488 0.9866488 0.7599813 0.7395237 1.2054476 0.7173943 2.2299485 0.7599813
209 210 211 212 213 214 215 216
1.4609860 2.2299485 2.2299485 0.9494681 1.2426320 0.7395237 0.9210563 0.7834244
217 218 219 220 221 222 223 224
0.7834244 0.9494681 0.9571244 2.2987358 1.5181975 2.2987358 2.2299485 0.7834244
225 226 227 228 229 230 231 232
1.5181975 1.2054476 1.5181975 0.8141029 2.2987358 0.9866488 2.2299485 1.2054476
233 234 235 236 237 238 239 240
2.2299485 0.9494681 0.7395237 0.9866488 0.7173943 2.2299485 1.5650294 2.2299485
241 242 243 244 245 246 247 248
1.2426320 2.2299485 2.2987358 0.9866488 2.2299485 0.7684831 0.7599813 2.2299485
249 250 251 252 253 254 255 256
2.2299485 2.2987358 2.2299485 2.2987358 1.2426320 1.2054476 0.7834244 0.9571244
257 258 259 260 261 262 263 264
0.7599813 2.2987358 2.2987358 1.2054476 0.9571244 2.2987358 1.2426320 0.7599813
265 266 267 268 269 270 271 272
0.7897418 2.2987358 2.2299485 1.5181975 2.2987358 0.9571244 2.2987358 0.7834244
273 274 275 276 277 278 279 280
0.6837339 1.5650294 0.9210563 1.2054476 0.9866488 2.2299485 2.2121107 0.7834244
281 282 283 284 285 286 287 288
2.2987358 2.2299485 1.2054476 2.2299485 1.2054476 0.7834244 0.7834244 0.9866488
289 290 291 292 293 294 295 296
1.2054476 2.2299485 1.2054476 0.7834244 2.2987358 0.7599813 1.5181975 0.9494681
297 298 299 300 301 302 303 304
0.6992588 0.9210563 2.2987358 1.5650294 1.2054476 0.9866488 0.9866488 0.7834244
305 306 307 308 309 310 311 312
0.9571244 0.7173943 1.5650294 0.7173943 2.2987358 0.9494681 0.7599813 1.2054476
313 314 315 316 317 318 319 320
2.2987358 0.9866488 2.2299485 0.9866488 0.9571244 2.2987358 0.7599813 2.2987358
321 322 323 324 325 326 327 328
0.6992588 0.7599813 0.9866488 2.2299485 0.7599813 0.9494681 2.2987358 1.4609860
329 330 331 332 333 334 335 336
1.5181975 0.8141029 0.7834244 1.5181975 2.2299485 1.5181975 0.9866488 2.2299485
337 338 339 340 341 342 343 344
0.7173943 2.2299485 2.2299485 1.5181975 1.1958049 2.2299485 0.8141029 0.7454871
345 346 347 348 349 350 351 352
0.9866488 0.7048250 0.8141029 1.2426320 2.2299485 0.7834244 0.7599813 0.7599813
353 354 355 356 357 358 359 360
0.7048250 0.9571244 0.9571244 2.2299485 2.2299485 1.2426320 0.7395237 0.7048250
361 362 363 364 365 366 367 368
1.5650294 0.6992588 2.2299485 1.5181975 2.2987358 0.9866488 1.2054476 1.5181975
369 370 371 372 373 374 375 376
1.5181975 0.9210563 1.2054476 2.2987358 1.2426320 0.7599813 0.7834244 0.7897418
377 378 379 380 381 382 383 384
1.2054476 2.2299485 0.7173943 0.7897418 0.9571244 2.2987358 1.5650294 2.2987358
385 386 387 388 389 390 391 392
1.2054476 0.7599813 1.5181975 0.7173943 0.7599813 2.2299485 1.2054476 2.2987358
393 394 395 396 397 398 399 400
2.2299485 2.2987358 1.2054476 0.9494681 0.7599813 0.7173943 2.2299485 0.7834244
401 402 403 404 405 406 407 408
0.9866488 0.9571244 2.2987358 0.7834244 2.2121107 1.1958049 2.2987358 2.2987358
409 410 411 412 413 414 415 416
1.1600217 1.2054476 0.9210563 1.2426320 1.1958049 1.5650294 0.7599813 0.9571244
417 418 419 420 421 422 423 424
2.1459156 2.2987358 1.2054476 1.4609860 0.9866488 0.9571244 2.2299485 0.7599813
425 426 427 428 429 430 431 432
1.2426320 0.9866488 0.7048250 1.2054476 2.2299485 1.5650294 2.2299485 0.7599813
433 434 435 436 437 438 439 440
0.9866488 2.2299485 1.2054476 1.2054476 2.2299485 2.2299485 1.2426320 1.2426320
441 442 443 444 445 446 447 448
2.2299485 0.9866488 1.2054476 0.7599813 0.9571244 0.9210563 0.7599813 2.2121107
449 450 451 452 453 454 455 456
1.5650294 2.2987358 0.9571244 0.7834244 0.9571244 2.2987358 1.5181975 0.7048250
457 458 459 460 461 462 463 464
0.7897418 2.2987358 0.7599813 2.2987358 2.2299485 2.2299485 2.2299485 2.2299485
465 466 467 468 469 470 471 472
1.5650294 2.2299485 0.9494681 1.5650294 0.9210563 2.2987358 2.2299485 0.9571244
473 474 475 476 477 478 479 480
1.2054476 0.8141029 1.2054476 2.2299485 1.5181975 0.7834244 0.9494681 1.2054476
481 482 483 484 485 486 487 488
1.5181975 0.9866488 0.7173943 1.5181975 1.2054476 2.2987358 1.4609860 1.5181975
489 490 491 492 493 494 495 496
2.2987358 2.2299485 2.2987358 2.2987358 0.9210563 1.2426320 0.9866488 2.2987358
497 498 499 500 501 502 503 504
2.2299485 0.9494681 1.2054476 0.7490561 2.2987358 2.2987358 0.9866488 0.9494681
505 506 507 508 509 510 511 512
2.2299485 0.9494681 1.5181975 1.2426320 1.2426320 0.7834244 0.9866488 2.2987358
513 514 515 516 517 518 519 520
2.2299485 0.9571244 0.9866488 1.2426320 2.2987358 2.2299485 1.5650294 2.2987358
521 522 523 524 525 526 527 528
2.2299485 1.2054476 1.2426320 0.9494681 1.2426320 2.2299485 2.2299485 1.1600217
529 530 531 532 533 534 535 536
1.2054476 0.8141029 2.2299485 1.5650294 0.9866488 0.9571244 2.2299485 1.5650294
537 538 539 540 541 542 543 544
1.5650294 1.2054476 1.2054476 1.5650294 0.7454871 2.2299485 1.1600217 1.2426320
545 546 547 548 549 550 551 552
0.9866488 0.7897418 2.1459156 2.2299485 2.2987358 1.5060531 2.2299485 2.2987358
553 554 555 556 557 558 559 560
2.2987358 2.2299485 2.2987358 2.2987358 1.5650294 2.2987358 2.2299485 1.5650294
561 562 563 564 565 566 567 568
2.2987358 0.7599813 2.2299485 2.2987358 0.9210563 1.5181975 0.9571244 1.1958049
569 570 571 572 573 574 575 576
0.9571244 2.2299485 0.9210563 0.7454871 2.2299485 0.9210563 0.9866488 0.7897418
577 578 579 580 581 582 583 584
2.2299485 1.2426320 2.2121107 2.2987358 0.9866488 2.2987358 2.2987358 0.7834244
585 586 587 588 589 590 591 592
0.9866488 2.2987358 2.2299485 2.2121107 2.2299485 2.2299485 1.2054476 2.2987358
593 594 595 596 597 598 599 600
2.2299485 1.2426320 1.5650294 2.2987358 0.8141029 1.5181975 1.5650294 1.2426320
601
2.2299485 Fitted values for the original data
Simple data frame with broom naming convention:
rowid type predicted std.error conf.low conf.high affairs children
1 1 response 1.5060531 0.14741307 1.2431523 1.8245519 0 no
2 2 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
3 3 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
4 4 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
5 5 response 0.7395237 0.06835445 0.6169859 0.8863984 0 no
6 6 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
7 7 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
8 8 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
9 9 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
10 10 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
11 11 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
12 12 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
13 13 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
14 14 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
15 15 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
16 16 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
17 17 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
18 18 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
19 19 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
20 20 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
21 21 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
22 22 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
23 23 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
24 24 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
25 25 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
26 26 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
27 27 response 1.1600217 0.10233572 0.9758300 1.3789804 0 no
28 28 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
29 29 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
30 30 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
31 31 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
32 32 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
33 33 response 0.7208288 0.06721263 0.6004310 0.8653687 0 no
34 34 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
35 35 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
36 36 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
37 37 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
38 38 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
39 39 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
40 40 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
41 41 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
42 42 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
43 43 response 0.6992588 0.06369030 0.5849359 0.8359255 0 no
44 44 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
45 45 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
46 46 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
47 47 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
48 48 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
49 49 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
50 50 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
51 51 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
52 52 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
53 53 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
54 54 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
55 55 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
56 56 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
57 57 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
58 58 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
59 59 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
60 60 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
61 61 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
62 62 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
63 63 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
64 64 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
65 65 response 1.1958049 0.10787230 1.0020153 1.4270735 0 no
66 66 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
67 67 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
68 68 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
69 69 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
70 70 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
71 71 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
72 72 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
73 73 response 0.6837339 0.06280417 0.5710843 0.8186042 0 no
74 74 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
75 75 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
76 76 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
77 77 response 0.7897418 0.07416668 0.6569716 0.9493440 0 yes
78 78 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
79 79 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
80 80 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
81 81 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
82 82 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
83 83 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
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85 85 response 0.7897418 0.07416668 0.6569716 0.9493440 0 yes
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87 87 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
88 88 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
89 89 response 0.7897418 0.07416668 0.6569716 0.9493440 0 yes
90 90 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
91 91 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
92 92 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
93 93 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
94 94 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
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96 96 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
97 97 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
98 98 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
99 99 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
100 100 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
101 101 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
102 102 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
103 103 response 0.6837339 0.06280417 0.5710843 0.8186042 0 no
104 104 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
105 105 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
106 106 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
107 107 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
108 108 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
109 109 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
110 110 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
111 111 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
112 112 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
113 113 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
114 114 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
115 115 response 1.1600217 0.10233572 0.9758300 1.3789804 0 no
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117 117 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
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119 119 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
120 120 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
121 121 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
122 122 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
123 123 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
124 124 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
125 125 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
126 126 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
127 127 response 0.6837339 0.06280417 0.5710843 0.8186042 0 no
128 128 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
129 129 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
130 130 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
131 131 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
132 132 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
133 133 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
134 134 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
135 135 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
136 136 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
137 137 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
138 138 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
139 139 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
140 140 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
141 141 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
142 142 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
143 143 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
144 144 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
145 145 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
146 146 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
147 147 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
148 148 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
149 149 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
150 150 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
151 151 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
152 152 response 0.7395237 0.06835445 0.6169859 0.8863984 0 no
153 153 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
154 154 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
155 155 response 0.7684831 0.07638978 0.6324440 0.9337844 0 yes
156 156 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
157 157 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
158 158 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
159 159 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
160 160 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
161 161 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
162 162 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
163 163 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
164 164 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
165 165 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
166 166 response 0.7897418 0.07416668 0.6569716 0.9493440 0 yes
167 167 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
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169 169 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
170 170 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
171 171 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
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173 173 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
174 174 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
175 175 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
176 176 response 0.7324256 0.07585460 0.5978713 0.8972620 0 yes
177 177 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
178 178 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
179 179 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
180 180 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
181 181 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
182 182 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
183 183 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
184 184 response 1.1600217 0.10233572 0.9758300 1.3789804 0 no
185 185 response 0.7266414 0.07342315 0.5960887 0.8857872 0 yes
186 186 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
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188 188 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
189 189 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
190 190 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
191 191 response 0.7395237 0.06835445 0.6169859 0.8863984 0 no
192 192 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
193 193 response 0.7490561 0.07611485 0.6137904 0.9141314 0 yes
194 194 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
195 195 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
196 196 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
197 197 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
198 198 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
199 199 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
200 200 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
201 201 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
202 202 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
203 203 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
204 204 response 0.7395237 0.06835445 0.6169859 0.8863984 0 no
205 205 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
206 206 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
207 207 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
208 208 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
209 209 response 1.4609860 0.14047645 1.2100449 1.7639677 0 no
210 210 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
211 211 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
212 212 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
213 213 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
214 214 response 0.7395237 0.06835445 0.6169859 0.8863984 0 no
215 215 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
216 216 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
217 217 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
218 218 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
219 219 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
220 220 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
221 221 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
222 222 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
223 223 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
224 224 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
225 225 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
226 226 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
227 227 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
228 228 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
229 229 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
230 230 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
231 231 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
232 232 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
233 233 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
234 234 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
235 235 response 0.7395237 0.06835445 0.6169859 0.8863984 0 no
236 236 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
237 237 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
238 238 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
239 239 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
240 240 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
241 241 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
242 242 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
243 243 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
244 244 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
245 245 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
246 246 response 0.7684831 0.07638978 0.6324440 0.9337844 0 yes
247 247 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
248 248 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
249 249 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
250 250 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
251 251 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
252 252 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
253 253 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
254 254 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
255 255 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
256 256 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
257 257 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
258 258 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
259 259 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
260 260 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
261 261 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
262 262 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
263 263 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
264 264 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
265 265 response 0.7897418 0.07416668 0.6569716 0.9493440 0 yes
266 266 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
267 267 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
268 268 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
269 269 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
270 270 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
271 271 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
272 272 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
273 273 response 0.6837339 0.06280417 0.5710843 0.8186042 0 no
274 274 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
275 275 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
276 276 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
277 277 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
278 278 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
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280 280 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
281 281 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
282 282 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
283 283 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
284 284 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
285 285 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
286 286 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
287 287 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
288 288 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
289 289 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
290 290 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
291 291 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
292 292 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
293 293 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
294 294 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
295 295 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
296 296 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
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298 298 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
299 299 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
300 300 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
301 301 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
302 302 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
303 303 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
304 304 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
305 305 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
306 306 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
307 307 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
308 308 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
309 309 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
310 310 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
311 311 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
312 312 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
313 313 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
314 314 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
315 315 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
316 316 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
317 317 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
318 318 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
319 319 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
320 320 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
321 321 response 0.6992588 0.06369030 0.5849359 0.8359255 0 no
322 322 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
323 323 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
324 324 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
325 325 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
326 326 response 0.9494681 0.08362332 0.7989356 1.1283634 0 no
327 327 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
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329 329 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
330 330 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
331 331 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
332 332 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
333 333 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
334 334 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
335 335 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
336 336 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
337 337 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
338 338 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
339 339 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
340 340 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
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342 342 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
343 343 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
344 344 response 0.7454871 0.07367786 0.6142063 0.9048278 0 yes
345 345 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
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347 347 response 0.8141029 0.07692276 0.6764741 0.9797323 0 yes
348 348 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
349 349 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
350 350 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
351 351 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
352 352 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
353 353 response 0.7048250 0.06625926 0.5862211 0.8474247 0 no
354 354 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
355 355 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
356 356 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
357 357 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
358 358 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
359 359 response 0.7395237 0.06835445 0.6169859 0.8863984 0 no
360 360 response 0.7048250 0.06625926 0.5862211 0.8474247 0 no
361 361 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
362 362 response 0.6992588 0.06369030 0.5849359 0.8359255 0 no
363 363 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
364 364 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
365 365 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
366 366 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
367 367 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
368 368 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
369 369 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
370 370 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
371 371 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
372 372 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
373 373 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
374 374 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
375 375 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
376 376 response 0.7897418 0.07416668 0.6569716 0.9493440 0 yes
377 377 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
378 378 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
379 379 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
380 380 response 0.7897418 0.07416668 0.6569716 0.9493440 0 yes
381 381 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
382 382 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
383 383 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
384 384 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
385 385 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
386 386 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
387 387 response 1.5181975 0.07908880 1.3708372 1.6813985 0 yes
388 388 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
389 389 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
390 390 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
391 391 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
392 392 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
393 393 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
394 394 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
395 395 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
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397 397 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
398 398 response 0.7173943 0.06475348 0.6010723 0.8562274 0 no
399 399 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
400 400 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
401 401 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
402 402 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
403 403 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
404 404 response 0.7834244 0.07116344 0.6556575 0.9360891 0 no
405 405 response 2.2121107 0.26495535 1.7492596 2.7974313 0 no
406 406 response 1.1958049 0.10787230 1.0020153 1.4270735 0 no
407 407 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
408 408 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
409 409 response 1.1600217 0.10233572 0.9758300 1.3789804 0 no
410 410 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
411 411 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
412 412 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
413 413 response 1.1958049 0.10787230 1.0020153 1.4270735 0 no
414 414 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
415 415 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
416 416 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
417 417 response 2.1459156 0.25430191 1.7011434 2.7069756 0 no
418 418 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
419 419 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
420 420 response 1.4609860 0.14047645 1.2100449 1.7639677 0 no
421 421 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
422 422 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
423 423 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
424 424 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
425 425 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
426 426 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
427 427 response 0.7048250 0.06625926 0.5862211 0.8474247 0 no
428 428 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
429 429 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
430 430 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
431 431 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
432 432 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
433 433 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
434 434 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
435 435 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
436 436 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
437 437 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
438 438 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
439 439 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
440 440 response 1.2426320 0.07826874 1.0983191 1.4059068 0 yes
441 441 response 2.2299485 0.12427095 1.9992124 2.4873147 0 yes
442 442 response 0.9866488 0.07787951 0.8452290 1.1517303 0 yes
443 443 response 1.2054476 0.07523205 1.0666570 1.3622972 0 yes
444 444 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
445 445 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
446 446 response 0.9210563 0.07913696 0.7783073 1.0899868 0 no
447 447 response 0.7599813 0.06737756 0.6387605 0.9042068 0 no
448 448 response 2.2121107 0.26495535 1.7492596 2.7974313 0 no
449 449 response 1.5650294 0.08231414 1.4117337 1.7349709 0 yes
450 450 response 2.2987358 0.12851060 2.0601684 2.5649293 0 yes
451 451 response 0.9571244 0.07498949 0.8208765 1.1159865 0 yes
452 452 response 0.7834244 0.07116344 0.6556575 0.9360891 3 no
453 453 response 0.9571244 0.07498949 0.8208765 1.1159865 3 yes
454 454 response 2.2987358 0.12851060 2.0601684 2.5649293 7 yes
455 455 response 1.5181975 0.07908880 1.3708372 1.6813985 12 yes
456 456 response 0.7048250 0.06625926 0.5862211 0.8474247 1 no
457 457 response 0.7897418 0.07416668 0.6569716 0.9493440 1 yes
458 458 response 2.2987358 0.12851060 2.0601684 2.5649293 12 yes
459 459 response 0.7599813 0.06737756 0.6387605 0.9042068 7 no
460 460 response 2.2987358 0.12851060 2.0601684 2.5649293 2 yes
461 461 response 2.2299485 0.12427095 1.9992124 2.4873147 3 yes
462 462 response 2.2299485 0.12427095 1.9992124 2.4873147 1 yes
463 463 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
464 464 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
465 465 response 1.5650294 0.08231414 1.4117337 1.7349709 12 yes
466 466 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
467 467 response 0.9494681 0.08362332 0.7989356 1.1283634 3 no
468 468 response 1.5650294 0.08231414 1.4117337 1.7349709 7 yes
469 469 response 0.9210563 0.07913696 0.7783073 1.0899868 7 no
470 470 response 2.2987358 0.12851060 2.0601684 2.5649293 1 yes
471 471 response 2.2299485 0.12427095 1.9992124 2.4873147 1 yes
472 472 response 0.9571244 0.07498949 0.8208765 1.1159865 7 yes
473 473 response 1.2054476 0.07523205 1.0666570 1.3622972 1 yes
474 474 response 0.8141029 0.07692276 0.6764741 0.9797323 12 yes
475 475 response 1.2054476 0.07523205 1.0666570 1.3622972 12 yes
476 476 response 2.2299485 0.12427095 1.9992124 2.4873147 3 yes
477 477 response 1.5181975 0.07908880 1.3708372 1.6813985 7 yes
478 478 response 0.7834244 0.07116344 0.6556575 0.9360891 1 no
479 479 response 0.9494681 0.08362332 0.7989356 1.1283634 1 no
480 480 response 1.2054476 0.07523205 1.0666570 1.3622972 1 yes
481 481 response 1.5181975 0.07908880 1.3708372 1.6813985 3 yes
482 482 response 0.9866488 0.07787951 0.8452290 1.1517303 3 yes
483 483 response 0.7173943 0.06475348 0.6010723 0.8562274 1 no
484 484 response 1.5181975 0.07908880 1.3708372 1.6813985 1 yes
485 485 response 1.2054476 0.07523205 1.0666570 1.3622972 7 yes
486 486 response 2.2987358 0.12851060 2.0601684 2.5649293 7 yes
487 487 response 1.4609860 0.14047645 1.2100449 1.7639677 7 no
488 488 response 1.5181975 0.07908880 1.3708372 1.6813985 12 yes
489 489 response 2.2987358 0.12851060 2.0601684 2.5649293 7 yes
490 490 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
491 491 response 2.2987358 0.12851060 2.0601684 2.5649293 1 yes
492 492 response 2.2987358 0.12851060 2.0601684 2.5649293 2 yes
493 493 response 0.9210563 0.07913696 0.7783073 1.0899868 12 no
494 494 response 1.2426320 0.07826874 1.0983191 1.4059068 12 yes
495 495 response 0.9866488 0.07787951 0.8452290 1.1517303 1 yes
496 496 response 2.2987358 0.12851060 2.0601684 2.5649293 12 yes
497 497 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
498 498 response 0.9494681 0.08362332 0.7989356 1.1283634 7 no
499 499 response 1.2054476 0.07523205 1.0666570 1.3622972 7 yes
500 500 response 0.7490561 0.07611485 0.6137904 0.9141314 1 yes
501 501 response 2.2987358 0.12851060 2.0601684 2.5649293 3 yes
502 502 response 2.2987358 0.12851060 2.0601684 2.5649293 12 yes
503 503 response 0.9866488 0.07787951 0.8452290 1.1517303 7 yes
504 504 response 0.9494681 0.08362332 0.7989356 1.1283634 1 no
505 505 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
506 506 response 0.9494681 0.08362332 0.7989356 1.1283634 1 no
507 507 response 1.5181975 0.07908880 1.3708372 1.6813985 1 yes
508 508 response 1.2426320 0.07826874 1.0983191 1.4059068 1 yes
509 509 response 1.2426320 0.07826874 1.0983191 1.4059068 7 yes
510 510 response 0.7834244 0.07116344 0.6556575 0.9360891 3 no
511 511 response 0.9866488 0.07787951 0.8452290 1.1517303 7 yes
512 512 response 2.2987358 0.12851060 2.0601684 2.5649293 7 yes
513 513 response 2.2299485 0.12427095 1.9992124 2.4873147 2 yes
514 514 response 0.9571244 0.07498949 0.8208765 1.1159865 7 yes
515 515 response 0.9866488 0.07787951 0.8452290 1.1517303 1 yes
516 516 response 1.2426320 0.07826874 1.0983191 1.4059068 7 yes
517 517 response 2.2987358 0.12851060 2.0601684 2.5649293 2 yes
518 518 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
519 519 response 1.5650294 0.08231414 1.4117337 1.7349709 7 yes
520 520 response 2.2987358 0.12851060 2.0601684 2.5649293 3 yes
521 521 response 2.2299485 0.12427095 1.9992124 2.4873147 1 yes
522 522 response 1.2054476 0.07523205 1.0666570 1.3622972 2 yes
523 523 response 1.2426320 0.07826874 1.0983191 1.4059068 12 yes
524 524 response 0.9494681 0.08362332 0.7989356 1.1283634 1 no
525 525 response 1.2426320 0.07826874 1.0983191 1.4059068 3 yes
526 526 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
527 527 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
528 528 response 1.1600217 0.10233572 0.9758300 1.3789804 7 no
529 529 response 1.2054476 0.07523205 1.0666570 1.3622972 1 yes
530 530 response 0.8141029 0.07692276 0.6764741 0.9797323 1 yes
531 531 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
532 532 response 1.5650294 0.08231414 1.4117337 1.7349709 7 yes
533 533 response 0.9866488 0.07787951 0.8452290 1.1517303 7 yes
534 534 response 0.9571244 0.07498949 0.8208765 1.1159865 1 yes
535 535 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
536 536 response 1.5650294 0.08231414 1.4117337 1.7349709 1 yes
537 537 response 1.5650294 0.08231414 1.4117337 1.7349709 12 yes
538 538 response 1.2054476 0.07523205 1.0666570 1.3622972 12 yes
539 539 response 1.2054476 0.07523205 1.0666570 1.3622972 3 yes
540 540 response 1.5650294 0.08231414 1.4117337 1.7349709 3 yes
541 541 response 0.7454871 0.07367786 0.6142063 0.9048278 12 yes
542 542 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
543 543 response 1.1600217 0.10233572 0.9758300 1.3789804 2 no
544 544 response 1.2426320 0.07826874 1.0983191 1.4059068 1 yes
545 545 response 0.9866488 0.07787951 0.8452290 1.1517303 7 yes
546 546 response 0.7897418 0.07416668 0.6569716 0.9493440 1 yes
547 547 response 2.1459156 0.25430191 1.7011434 2.7069756 12 no
548 548 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
549 549 response 2.2987358 0.12851060 2.0601684 2.5649293 7 yes
550 550 response 1.5060531 0.14741307 1.2431523 1.8245519 12 no
551 551 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
552 552 response 2.2987358 0.12851060 2.0601684 2.5649293 7 yes
553 553 response 2.2987358 0.12851060 2.0601684 2.5649293 12 yes
554 554 response 2.2299485 0.12427095 1.9992124 2.4873147 2 yes
555 555 response 2.2987358 0.12851060 2.0601684 2.5649293 12 yes
556 556 response 2.2987358 0.12851060 2.0601684 2.5649293 12 yes
557 557 response 1.5650294 0.08231414 1.4117337 1.7349709 7 yes
558 558 response 2.2987358 0.12851060 2.0601684 2.5649293 2 yes
559 559 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
560 560 response 1.5650294 0.08231414 1.4117337 1.7349709 7 yes
561 561 response 2.2987358 0.12851060 2.0601684 2.5649293 2 yes
562 562 response 0.7599813 0.06737756 0.6387605 0.9042068 7 no
563 563 response 2.2299485 0.12427095 1.9992124 2.4873147 3 yes
564 564 response 2.2987358 0.12851060 2.0601684 2.5649293 12 yes
565 565 response 0.9210563 0.07913696 0.7783073 1.0899868 12 no
566 566 response 1.5181975 0.07908880 1.3708372 1.6813985 2 yes
567 567 response 0.9571244 0.07498949 0.8208765 1.1159865 1 yes
568 568 response 1.1958049 0.10787230 1.0020153 1.4270735 12 no
569 569 response 0.9571244 0.07498949 0.8208765 1.1159865 2 yes
570 570 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
571 571 response 0.9210563 0.07913696 0.7783073 1.0899868 2 no
572 572 response 0.7454871 0.07367786 0.6142063 0.9048278 12 yes
573 573 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
574 574 response 0.9210563 0.07913696 0.7783073 1.0899868 7 no
575 575 response 0.9866488 0.07787951 0.8452290 1.1517303 2 yes
576 576 response 0.7897418 0.07416668 0.6569716 0.9493440 1 yes
577 577 response 2.2299485 0.12427095 1.9992124 2.4873147 3 yes
578 578 response 1.2426320 0.07826874 1.0983191 1.4059068 1 yes
579 579 response 2.2121107 0.26495535 1.7492596 2.7974313 12 no
580 580 response 2.2987358 0.12851060 2.0601684 2.5649293 1 yes
581 581 response 0.9866488 0.07787951 0.8452290 1.1517303 1 yes
582 582 response 2.2987358 0.12851060 2.0601684 2.5649293 2 yes
583 583 response 2.2987358 0.12851060 2.0601684 2.5649293 7 yes
584 584 response 0.7834244 0.07116344 0.6556575 0.9360891 3 no
585 585 response 0.9866488 0.07787951 0.8452290 1.1517303 3 yes
586 586 response 2.2987358 0.12851060 2.0601684 2.5649293 2 yes
587 587 response 2.2299485 0.12427095 1.9992124 2.4873147 12 yes
588 588 response 2.2121107 0.26495535 1.7492596 2.7974313 12 no
589 589 response 2.2299485 0.12427095 1.9992124 2.4873147 3 yes
590 590 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
591 591 response 1.2054476 0.07523205 1.0666570 1.3622972 7 yes
592 592 response 2.2987358 0.12851060 2.0601684 2.5649293 12 yes
593 593 response 2.2299485 0.12427095 1.9992124 2.4873147 7 yes
594 594 response 1.2426320 0.07826874 1.0983191 1.4059068 12 yes
595 595 response 1.5650294 0.08231414 1.4117337 1.7349709 3 yes
596 596 response 2.2987358 0.12851060 2.0601684 2.5649293 7 yes
597 597 response 0.8141029 0.07692276 0.6764741 0.9797323 1 yes
598 598 response 1.5181975 0.07908880 1.3708372 1.6813985 7 yes
599 599 response 1.5650294 0.08231414 1.4117337 1.7349709 2 yes
600 600 response 1.2426320 0.07826874 1.0983191 1.4059068 2 yes
601 601 response 2.2299485 0.12427095 1.9992124 2.4873147 1 yes
gender yearsmarried
1 male 10.000
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30 female 4.000
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272 male 1.500
273 female 0.125
274 male 10.000
275 female 4.000
276 female 7.000
277 male 4.000
278 female 15.000
279 male 15.000
280 male 1.500
281 male 15.000
282 female 15.000
283 female 7.000
284 female 15.000
285 female 7.000
286 male 1.500
287 male 1.500
288 male 4.000
289 female 7.000
290 female 15.000
291 female 7.000
292 male 1.500
293 male 15.000
294 female 1.500
295 female 10.000
296 male 4.000
297 female 0.417
298 female 4.000
299 male 15.000
300 male 10.000
301 female 7.000
302 male 4.000
303 male 4.000
304 male 1.500
305 female 4.000
306 female 0.750
307 male 10.000
308 female 0.750
309 male 15.000
310 male 4.000
311 female 1.500
312 female 7.000
313 male 15.000
314 male 4.000
315 female 15.000
316 male 4.000
317 female 4.000
318 male 15.000
319 female 1.500
320 male 15.000
321 female 0.417
322 female 1.500
323 male 4.000
324 female 15.000
325 female 1.500
326 male 4.000
327 male 15.000
328 female 10.000
329 female 10.000
330 male 1.500
331 male 1.500
332 female 10.000
333 female 15.000
334 female 10.000
335 male 4.000
336 female 15.000
337 female 0.750
338 female 15.000
339 female 15.000
340 female 10.000
341 male 7.000
342 female 15.000
343 male 1.500
344 female 0.750
345 male 4.000
346 male 0.125
347 male 1.500
348 male 7.000
349 female 15.000
350 male 1.500
351 female 1.500
352 female 1.500
353 male 0.125
354 female 4.000
355 female 4.000
356 female 15.000
357 female 15.000
358 male 7.000
359 male 0.750
360 male 0.125
361 male 10.000
362 female 0.417
363 female 15.000
364 female 10.000
365 male 15.000
366 male 4.000
367 female 7.000
368 female 10.000
369 female 10.000
370 female 4.000
371 female 7.000
372 male 15.000
373 male 7.000
374 female 1.500
375 male 1.500
376 female 1.500
377 female 7.000
378 female 15.000
379 female 0.750
380 female 1.500
381 female 4.000
382 male 15.000
383 male 10.000
384 male 15.000
385 female 7.000
386 female 1.500
387 female 10.000
388 female 0.750
389 female 1.500
390 female 15.000
391 female 7.000
392 male 15.000
393 female 15.000
394 male 15.000
395 female 7.000
396 male 4.000
397 female 1.500
398 female 0.750
399 female 15.000
400 male 1.500
401 male 4.000
402 female 4.000
403 male 15.000
404 male 1.500
405 male 15.000
406 male 7.000
407 male 15.000
408 male 15.000
409 female 7.000
410 female 7.000
411 female 4.000
412 male 7.000
413 male 7.000
414 male 10.000
415 female 1.500
416 female 4.000
417 female 15.000
418 male 15.000
419 female 7.000
420 female 10.000
421 male 4.000
422 female 4.000
423 female 15.000
424 female 1.500
425 male 7.000
426 male 4.000
427 male 0.125
428 female 7.000
429 female 15.000
430 male 10.000
431 female 15.000
432 female 1.500
433 male 4.000
434 female 15.000
435 female 7.000
436 female 7.000
437 female 15.000
438 female 15.000
439 male 7.000
440 male 7.000
441 female 15.000
442 male 4.000
443 female 7.000
444 female 1.500
445 female 4.000
446 female 4.000
447 female 1.500
448 male 15.000
449 male 10.000
450 male 15.000
451 female 4.000
452 male 1.500
453 female 4.000
454 male 15.000
455 female 10.000
456 male 0.125
457 female 1.500
458 male 15.000
459 female 1.500
460 male 15.000
461 female 15.000
462 female 15.000
463 female 15.000
464 female 15.000
465 male 10.000
466 female 15.000
467 male 4.000
468 male 10.000
469 female 4.000
470 male 15.000
471 female 15.000
472 female 4.000
473 female 7.000
474 male 1.500
475 female 7.000
476 female 15.000
477 female 10.000
478 male 1.500
479 male 4.000
480 female 7.000
481 female 10.000
482 male 4.000
483 female 0.750
484 female 10.000
485 female 7.000
486 male 15.000
487 female 10.000
488 female 10.000
489 male 15.000
490 female 15.000
491 male 15.000
492 male 15.000
493 female 4.000
494 male 7.000
495 male 4.000
496 male 15.000
497 female 15.000
498 male 4.000
499 female 7.000
500 male 0.417
501 male 15.000
502 male 15.000
503 male 4.000
504 male 4.000
505 female 15.000
506 male 4.000
507 female 10.000
508 male 7.000
509 male 7.000
510 male 1.500
511 male 4.000
512 male 15.000
513 female 15.000
514 female 4.000
515 male 4.000
516 male 7.000
517 male 15.000
518 female 15.000
519 male 10.000
520 male 15.000
521 female 15.000
522 female 7.000
523 male 7.000
524 male 4.000
525 male 7.000
526 female 15.000
527 female 15.000
528 female 7.000
529 female 7.000
530 male 1.500
531 female 15.000
532 male 10.000
533 male 4.000
534 female 4.000
535 female 15.000
536 male 10.000
537 male 10.000
538 female 7.000
539 female 7.000
540 male 10.000
541 female 0.750
542 female 15.000
543 female 7.000
544 male 7.000
545 male 4.000
546 female 1.500
547 female 15.000
548 female 15.000
549 male 15.000
550 male 10.000
551 female 15.000
552 male 15.000
553 male 15.000
554 female 15.000
555 male 15.000
556 male 15.000
557 male 10.000
558 male 15.000
559 female 15.000
560 male 10.000
561 male 15.000
562 female 1.500
563 female 15.000
564 male 15.000
565 female 4.000
566 female 10.000
567 female 4.000
568 male 7.000
569 female 4.000
570 female 15.000
571 female 4.000
572 female 0.750
573 female 15.000
574 female 4.000
575 male 4.000
576 female 1.500
577 female 15.000
578 male 7.000
579 male 15.000
580 male 15.000
581 male 4.000
582 male 15.000
583 male 15.000
584 male 1.500
585 male 4.000
586 male 15.000
587 female 15.000
588 male 15.000
589 female 15.000
590 female 15.000
591 female 7.000
592 male 15.000
593 female 15.000
594 male 7.000
595 male 10.000
596 male 15.000
597 male 1.500
598 female 10.000
599 male 10.000
600 male 7.000
601 female 15.000Predictions on a user-specified grid
Use the datagrid() function and newdata argument:
rowid type predicted std.error conf.low conf.high children gender
1 1 response 0.759956 0.07385389 0.6281555 0.9194111 yes female
2 2 response 4.810918 0.54304955 3.8560759 6.0021990 yes female
yearsmarried affairs
1 1 0
2 25 0Unspecified variables held at their mean or mode.
Predictions: hypothesis
Two predictions at the range of yearsmarried:
rowid type predicted std.error conf.low conf.high children gender
1 1 response 0.7105085 0.07318095 0.580627 0.8694434 yes female
2 2 response 2.2299485 0.12427095 1.999212 2.4873147 yes female
yearsmarried affairs
1 0.125 0
2 15.000 0Are they equal?
Conditional predictions
How does the outcome predicted by my model change as a function of the predictors?
Average predictions
Predictions on different scales
Response scale (expected counts)
rowid type predicted std.error conf.low conf.high affairs children
1 1 response 1.5060531 0.14741307 1.2431523 1.824552 0 no
2 2 response 0.9210563 0.07913696 0.7783073 1.089987 0 no
gender yearsmarried
1 male 10
2 female 4Link scale (log)
rowid type predicted std.error statistic p.value conf.low
1 1 link 0.40949238 0.09788039 4.1835996 2.869293e-05 0.2176503
2 2 link -0.08223412 0.08591978 -0.9571034 3.385150e-01 -0.2506338
conf.high affairs children gender yearsmarried
1 0.60133442 0 no male 10
2 0.08616556 0 no female 4Contrasts
The difference, ratio, or other function of two adjusted predictions, calculated for meaningfully different regressor values (e.g., College graduates vs. Others).
Contrasts are suuuper important
- What is the difference in predicted cholesterol levels between people in the treatment and control groups?
- What happens to predicted \(Y\) if \(X\) increases by one standard deviation?
- Is the estimated probability of conversion higher if the customer sees a blue shirt instead of a red one?
Manual contrasts
If children changes from “no” to “yes”, the expected number of affairs increases by…
# Copy the original data twice
dat_no <- dat_yes <- affairs
# Set different values for the predictor of interest
dat_no$children <- "no"
dat_yes$children <- "yes"
# Predictions
p_no <- predict(mod, newdata = dat_no, type = "response")
p_yes <- predict(mod, newdata = dat_yes, type = "response")
# Compute contrasts for each individual
p_yes - p_no 1 2 3 4 5 6 7
0.05897627 0.03606809 0.08403296 0.08662512 0.02895937 0.02976048 0.02809279
8 9 10 11 12 13 14
0.08662512 0.08403296 0.03067850 0.08662512 0.03718069 0.08662512 0.02976048
15 16 17 18 19 20 21
0.03606809 0.08403296 0.08403296 0.02809279 0.02976048 0.05721146 0.02976048
22 23 24 25 26 27 28
0.02976048 0.05721146 0.05721146 0.03718069 0.02976048 0.04542586 0.08662512
29 30 31 32 33 34 35
0.03718069 0.03606809 0.08662512 0.02976048 0.02822729 0.08403296 0.03718069
36 37 38 39 40 41 42
0.02976048 0.08403296 0.03606809 0.03067850 0.02809279 0.05897627 0.08662512
43 44 45 46 47 48 49
0.02738262 0.03606809 0.04542586 0.03718069 0.04542586 0.08403296 0.03067850
50 51 52 53 54 55 56
0.08662512 0.02809279 0.04682711 0.03718069 0.05897627 0.03718069 0.08403296
57 58 59 60 61 62 63
0.03067850 0.08403296 0.02809279 0.05897627 0.08403296 0.05897627 0.02809279
64 65 66 67 68 69 70
0.03718069 0.04682711 0.08662512 0.05897627 0.08403296 0.02976048 0.02976048
71 72 73 74 75 76 77
0.03606809 0.05721146 0.02677467 0.08662512 0.08662512 0.04682711 0.02976048
78 79 80 81 82 83 84
0.03718069 0.02976048 0.08662512 0.03067850 0.08662512 0.02809279 0.03718069
85 86 87 88 89 90 91
0.02976048 0.02738262 0.08403296 0.02976048 0.02976048 0.08403296 0.08403296
92 93 94 95 96 97 98
0.05897627 0.05897627 0.08662512 0.02822729 0.08403296 0.08662512 0.05897627
99 100 101 102 103 104 105
0.08662512 0.08662512 0.05721146 0.05897627 0.02677467 0.08662512 0.08403296
106 107 108 109 110 111 112
0.03718069 0.04682711 0.08662512 0.08662512 0.02976048 0.04542586 0.08403296
113 114 115 116 117 118 119
0.02976048 0.05897627 0.04542586 0.08662512 0.08662512 0.05897627 0.02809279
120 121 122 123 124 125 126
0.04542586 0.04542586 0.08403296 0.08662512 0.04542586 0.02976048 0.08662512
127 128 129 130 131 132 133
0.02677467 0.03067850 0.03067850 0.03067850 0.05721146 0.08662512 0.02976048
134 135 136 137 138 139 140
0.03718069 0.05721146 0.08403296 0.02809279 0.08403296 0.03718069 0.03718069
141 142 143 144 145 146 147
0.05721146 0.08403296 0.04682711 0.08662512 0.03718069 0.03718069 0.03718069
148 149 150 151 152 153 154
0.08403296 0.08403296 0.04682711 0.04682711 0.02895937 0.03718069 0.08662512
155 156 157 158 159 160 161
0.02895937 0.08662512 0.08403296 0.08403296 0.04682711 0.03718069 0.03606809
162 163 164 165 166 167 168
0.03606809 0.08662512 0.08403296 0.05721146 0.02976048 0.08403296 0.02976048
169 170 171 172 173 174 175
0.08662512 0.08662512 0.08662512 0.02677467 0.05721146 0.08403296 0.02976048
176 177 178 179 180 181 182
0.02760059 0.03606809 0.05721146 0.04542586 0.08403296 0.08403296 0.03067850
183 184 185 186 187 188 189
0.03606809 0.04542586 0.02738262 0.04542586 0.02895937 0.03718069 0.05897627
190 191 192 193 194 195 196
0.08662512 0.02895937 0.04542586 0.02822729 0.08662512 0.08403296 0.03718069
197 198 199 200 201 202 203
0.08662512 0.02976048 0.08662512 0.02976048 0.03718069 0.03718069 0.02976048
204 205 206 207 208 209 210
0.02895937 0.04542586 0.02809279 0.08403296 0.02976048 0.05721146 0.08403296
211 212 213 214 215 216 217
0.08403296 0.03718069 0.04682711 0.02895937 0.03606809 0.03067850 0.03067850
218 219 220 221 222 223 224
0.03718069 0.03606809 0.08662512 0.05721146 0.08662512 0.08403296 0.03067850
225 226 227 228 229 230 231
0.05721146 0.04542586 0.05721146 0.03067850 0.08662512 0.03718069 0.08403296
232 233 234 235 236 237 238
0.04542586 0.08403296 0.03718069 0.02895937 0.03718069 0.02809279 0.08403296
239 240 241 242 243 244 245
0.05897627 0.08403296 0.04682711 0.08403296 0.08662512 0.03718069 0.08403296
246 247 248 249 250 251 252
0.02895937 0.02976048 0.08403296 0.08403296 0.08662512 0.08403296 0.08662512
253 254 255 256 257 258 259
0.04682711 0.04542586 0.03067850 0.03606809 0.02976048 0.08662512 0.08662512
260 261 262 263 264 265 266
0.04542586 0.03606809 0.08662512 0.04682711 0.02976048 0.02976048 0.08662512
267 268 269 270 271 272 273
0.08403296 0.05721146 0.08662512 0.03606809 0.08662512 0.03067850 0.02677467
274 275 276 277 278 279 280
0.05897627 0.03606809 0.04542586 0.03718069 0.08403296 0.08662512 0.03067850
281 282 283 284 285 286 287
0.08662512 0.08403296 0.04542586 0.08403296 0.04542586 0.03067850 0.03067850
288 289 290 291 292 293 294
0.03718069 0.04542586 0.08403296 0.04542586 0.03067850 0.08662512 0.02976048
295 296 297 298 299 300 301
0.05721146 0.03718069 0.02738262 0.03606809 0.08662512 0.05897627 0.04542586
302 303 304 305 306 307 308
0.03718069 0.03718069 0.03067850 0.03606809 0.02809279 0.05897627 0.02809279
309 310 311 312 313 314 315
0.08662512 0.03718069 0.02976048 0.04542586 0.08662512 0.03718069 0.08403296
316 317 318 319 320 321 322
0.03718069 0.03606809 0.08662512 0.02976048 0.08662512 0.02738262 0.02976048
323 324 325 326 327 328 329
0.03718069 0.08403296 0.02976048 0.03718069 0.08662512 0.05721146 0.05721146
330 331 332 333 334 335 336
0.03067850 0.03067850 0.05721146 0.08403296 0.05721146 0.03718069 0.08403296
337 338 339 340 341 342 343
0.02809279 0.08403296 0.08403296 0.05721146 0.04682711 0.08403296 0.03067850
344 345 346 347 348 349 350
0.02809279 0.03718069 0.02760059 0.03067850 0.04682711 0.08403296 0.03067850
351 352 353 354 355 356 357
0.02976048 0.02976048 0.02760059 0.03606809 0.03606809 0.08403296 0.08403296
358 359 360 361 362 363 364
0.04682711 0.02895937 0.02760059 0.05897627 0.02738262 0.08403296 0.05721146
365 366 367 368 369 370 371
0.08662512 0.03718069 0.04542586 0.05721146 0.05721146 0.03606809 0.04542586
372 373 374 375 376 377 378
0.08662512 0.04682711 0.02976048 0.03067850 0.02976048 0.04542586 0.08403296
379 380 381 382 383 384 385
0.02809279 0.02976048 0.03606809 0.08662512 0.05897627 0.08662512 0.04542586
386 387 388 389 390 391 392
0.02976048 0.05721146 0.02809279 0.02976048 0.08403296 0.04542586 0.08662512
393 394 395 396 397 398 399
0.08403296 0.08662512 0.04542586 0.03718069 0.02976048 0.02809279 0.08403296
400 401 402 403 404 405 406
0.03067850 0.03718069 0.03606809 0.08662512 0.03067850 0.08662512 0.04682711
407 408 409 410 411 412 413
0.08662512 0.08662512 0.04542586 0.04542586 0.03606809 0.04682711 0.04682711
414 415 416 417 418 419 420
0.05897627 0.02976048 0.03606809 0.08403296 0.08662512 0.04542586 0.05721146
421 422 423 424 425 426 427
0.03718069 0.03606809 0.08403296 0.02976048 0.04682711 0.03718069 0.02760059
428 429 430 431 432 433 434
0.04542586 0.08403296 0.05897627 0.08403296 0.02976048 0.03718069 0.08403296
435 436 437 438 439 440 441
0.04542586 0.04542586 0.08403296 0.08403296 0.04682711 0.04682711 0.08403296
442 443 444 445 446 447 448
0.03718069 0.04542586 0.02976048 0.03606809 0.03606809 0.02976048 0.08662512
449 450 451 452 453 454 455
0.05897627 0.08662512 0.03606809 0.03067850 0.03606809 0.08662512 0.05721146
456 457 458 459 460 461 462
0.02760059 0.02976048 0.08662512 0.02976048 0.08662512 0.08403296 0.08403296
463 464 465 466 467 468 469
0.08403296 0.08403296 0.05897627 0.08403296 0.03718069 0.05897627 0.03606809
470 471 472 473 474 475 476
0.08662512 0.08403296 0.03606809 0.04542586 0.03067850 0.04542586 0.08403296
477 478 479 480 481 482 483
0.05721146 0.03067850 0.03718069 0.04542586 0.05721146 0.03718069 0.02809279
484 485 486 487 488 489 490
0.05721146 0.04542586 0.08662512 0.05721146 0.05721146 0.08662512 0.08403296
491 492 493 494 495 496 497
0.08662512 0.08662512 0.03606809 0.04682711 0.03718069 0.08662512 0.08403296
498 499 500 501 502 503 504
0.03718069 0.04542586 0.02822729 0.08662512 0.08662512 0.03718069 0.03718069
505 506 507 508 509 510 511
0.08403296 0.03718069 0.05721146 0.04682711 0.04682711 0.03067850 0.03718069
512 513 514 515 516 517 518
0.08662512 0.08403296 0.03606809 0.03718069 0.04682711 0.08662512 0.08403296
519 520 521 522 523 524 525
0.05897627 0.08662512 0.08403296 0.04542586 0.04682711 0.03718069 0.04682711
526 527 528 529 530 531 532
0.08403296 0.08403296 0.04542586 0.04542586 0.03067850 0.08403296 0.05897627
533 534 535 536 537 538 539
0.03718069 0.03606809 0.08403296 0.05897627 0.05897627 0.04542586 0.04542586
540 541 542 543 544 545 546
0.05897627 0.02809279 0.08403296 0.04542586 0.04682711 0.03718069 0.02976048
547 548 549 550 551 552 553
0.08403296 0.08403296 0.08662512 0.05897627 0.08403296 0.08662512 0.08662512
554 555 556 557 558 559 560
0.08403296 0.08662512 0.08662512 0.05897627 0.08662512 0.08403296 0.05897627
561 562 563 564 565 566 567
0.08662512 0.02976048 0.08403296 0.08662512 0.03606809 0.05721146 0.03606809
568 569 570 571 572 573 574
0.04682711 0.03606809 0.08403296 0.03606809 0.02809279 0.08403296 0.03606809
575 576 577 578 579 580 581
0.03718069 0.02976048 0.08403296 0.04682711 0.08662512 0.08662512 0.03718069
582 583 584 585 586 587 588
0.08662512 0.08662512 0.03067850 0.03718069 0.08662512 0.08403296 0.08662512
589 590 591 592 593 594 595
0.08403296 0.08403296 0.04542586 0.08662512 0.08403296 0.04682711 0.05897627
596 597 598 599 600 601
0.08662512 0.03067850 0.05721146 0.05897627 0.04682711 0.08403296 Average contrast
dat_no <- dat_yes <- affairs
dat_no$children <- "no"
dat_yes$children <- "yes"
p_no <- predict(mod, newdata = dat_no, type = "response")
p_yes <- predict(mod, newdata = dat_yes, type = "response")
mean(p_yes - p_no)[1] 0.05524715With marginaleffects:
Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 %
1 children yes - no 0.05525 0.14967 0.3691 0.71204 -0.23811
2 gender male - female 0.04425 0.09860 0.4488 0.65357 -0.14900
3 yearsmarried (x + 1) - x 0.11197 0.01183 9.4635 < 2e-16 0.08878
97.5 %
1 0.3486
2 0.2375
3 0.1352
Model type: glm
Prediction type: response How many additional affairs when yearsmarried increases from 1 to 25?
Term Contrast Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
1 yearsmarried 25 - 1 4.066 0.6327 6.427 1.3044e-10 2.826 5.307
Model type: glm
Prediction type: response How many additional affairs when yearsmarried increases by 1 SD?
Differences, Ratios, and More…
So far:
\[\hat{Y}_1 - \hat{Y}_0\]
Also:
\[e^{\hat{Y}_1 - \hat{Y}_0}\]
\[\frac{\hat{Y}_1}{\hat{Y}_0}\]
\[ln\left (\frac{\hat{Y}_1}{\hat{Y}_0} \right )\]
\[...\]
Marginal effects
A partial derivative (slope) of the regression equation with respect to a regressor of interest.
Terminology disaster
- “Marginal” = effect of small change (derivative)
- “Marginal” = sum or average of unit-level estimates (integral)
Marginal effects
In the simplest linear models, the marginal effect is just the coefficient:
\[y = \alpha + \beta x + \varepsilon\]
\[\frac{\partial y}{\partial x} = \beta\]
In more complicated cases, we need numerical derivatives.
General purpose technology: The slope/derivative measures the “effect” of a small change in \(X\) on \(Y\).
Marginal effects
Linear model with interaction:
Term Effect Std. Error z value Pr(>|z|) 2.5 % 97.5 %
1 hp -0.03051 0.007503 -4.066 4.7815e-05 -0.04521 -0.0158
2 wt -4.13165 0.529558 -7.802 6.0900e-15 -5.16956 -3.0937
Model type: lm
Prediction type: response Elasticity:
Conditional marginal effects plot
Interaction means that the “effect” of hp depends on the value of wt:
Marginal Means
Adjusted predictions of a model, averaged across a “reference grid” of categorical predictors.
Skipping this. Ask me about it in Q&A.
Weird model?
It Just Works™
Bayesian Random Effects
Andrew Heiss:
- Effect of “Autonomy of Opposition Parties” on “Freedom of the Press”
library(brms)
library(ggplot2)
library(ggdist)
vdem_2015 <- read.csv("https://github.com/vincentarelbundock/marginaleffects/raw/main/data-raw/vdem_2015.csv")
mod <- brm(
bf(media_index ~ party_autonomy + civil_liberties + (1 | region),
phi ~ (1 | region)),
data = vdem_2015,
family = Beta(),
control = list(adapt_delta = 0.9))Posterior predictions
region Predicted CI low CI high
1 Asia and Pacific 0.5915 0.5382 0.6438
2 Eastern Europe and Central Asia 0.6412 0.5993 0.6796
3 Latin America and the Caribbean 0.7517 0.7118 0.7867
4 Middle East and North Africa 0.4407 0.3760 0.5107
5 Sub-Saharan Africa 0.6525 0.6179 0.6837
6 Western Europe and North America 0.9212 0.8986 0.9368
Model type: brmsfit
Prediction type: response Draws from the posterior
Generalized Additive Models
Robust Standard Errors
Predictions, contrasts or marginal effects classical or heteroskedasticity or cluster-robust standard errors:
Tables with modelsummary
| OLS | Logit | |
|---|---|---|
| (Intercept) | -1.911 | -33.605 |
| (0.550) | (15.077) | |
| mpg | 0.086 | 1.260 |
| (0.018) | (0.567) | |
| hp | 0.004 | 0.055 |
| (0.002) | (0.027) | |
| Num.Obs. | 32 | 32 |
| R2 | 0.484 | |
| R2 Adj. | 0.449 | |
| AIC | 32.1 | 25.2 |
| BIC | 38.0 | 29.6 |
| Log.Lik. | -12.061 | -9.616 |
| F | 13.610 | 2.467 |
| RMSE | 0.35 | 0.32 |
Tables with modelsummary
| OLS | Logit | |
|---|---|---|
| mpg | 0.086 | 0.124 |
| (0.018) | (0.028) | |
| hp | 0.004 | 0.005 |
| (0.002) | (0.002) | |
| Num.Obs. | 32 | 32 |
| R2 | 0.484 | |
| R2 Adj. | 0.449 | |
| AIC | 32.1 | 25.2 |
| BIC | 38.0 | 29.6 |
| Log.Lik. | -12.061 | -9.616 |
| F | 13.610 | 2.467 |
| RMSE | 0.35 | 0.32 |
modelsummary
20,000+ words docs & case studies
Thanks to…
- Dependencies:
Rdata.table: ultra fast and powerfulcheckmate: user-input checksinsight: standardized “extractors” for any model
- Contributors:
- Marco Mendoza Aviña, Marcio Augusto Diniz, Resul Umit, Noah Greifer, Philippe Joly, Andrew Heiss, Tyson Barrett, Grant McDermott
- You!!
https://vincentarelbundock.github.io/marginaleffects